• Consider 3 methods of acoustic localisation

Medical diagnostic systems

Fundamental concepts in acoustics

(Orvosbiológiai képalkotó rendszerek)

(Alapfogalmak az akusztikában)

Miklós Gyöngy

Aims

• Consider 3 methods of acoustic localisation

• Through these, learn about concepts in acoustics

- propagation of sound - diffraction

- reflection - scattering - attenuation

• Link back to diagnostic ultrasound throughout

2. Binaural hearing passive

(difference in arrival times) 1. Lightning localisation

passive

(light as reference)

Methods of sound localisation

3. Echolocation active

(pulse-echo)

≈ 1 km every 3 s 343 m/s

• Passive method (with light as reference)

• Time of arrival (ToA), speed of sound (SoS) → localisation

• Analogy with diagnostic ultrasound?

• Where does speed of sound come from?

• What about propagation in tissue?

1. Lightning localisation

Analogy with diagnostic ultrasound:

localising “flashes of lightning” – photoacoustics

• Transmit laser pulse at known time

• Optically “dark” tissue absorbs laser preferentially

• Localised heating due to laser pulse creates shock wave

• Time of arrival depends on location of emission site

http://www.ucl.ac.uk/cabi/Photoacustics/Photoacustics.html

Propagation of sound

• Mechanical vibrations cause travelling waves

• Wave can be sustained by normal stress, shear stress, and volumetric compressions

• E→∞ or ρ→0: c→∞ (block moves as one)

ρ 0

c = C

propagation speed

elastic modulus

undisturbed density

Types of elastic moduli

• Young’s (E): axial propagation along laterally unconstrained rod

• P-wave (M): longitudal propagation, no lateral motion

• shear (G): motion transverse to direction of propagation

• bulk (K): volumetric propagation (pressure waves)

• K=M-4G/3: without shear, equivalent to P-wave

• K = -V ∂p/∂V : inverse of compressibility κ

• C = E (Young’s Modulus)

• E = σ/ε (stress/strain)

• c

= √(216×10

/7800) ≈ 5300 m/s

• E→∞ or ρ→0: c→∞ (block moves as one)

ρ

c = C

axial propagation

• ε

= ε

= 0

• no diffraction (high frequency)

• C = M (P-wave modulus)

• M = σ

/ε

(stress/strain)

• c

= 5980 m/s

ρ

c = C

longitudal propagation in bulk medium

• C = G (shear modulus)

• G = τ/γ (shear stress/shear strain)

• c

= √(84×10

/7800) ≈ 3300 m/s

ρ

c = C

transverse propagation

• C = K = 1/κ (bulk modulus=1/compressibility)

• K = -V ∂p/∂V

• K = M-4G/3

• K

: 2.05 GPa at 1 atm 3.88 GPa at 300 atm

• c

=√(K/ρ) ≈ √(2e6) ≈ 1400 m/s

ρ

c = C

volumetric propagation

Propagation of pressure waves

Assuming small pressure and density fluctuations

P = p₀ + p where p<<p R = ρ₀ + ρ where ρ<<ρ₀

• a waveform retains its shape as it travels (linear propagation)

• propagation can be described by the linear wave equation

Solutions of linear wave equation:

• planar wave propagating in z-direction: p=A g(z-t/c)

• spherical wave: p=A/|r-r₀| g(z-t/c)

1 0

2 2 2

2

=

∂

− ∂

∇ t

p

p c

Energetics of pressure waves

• A wave causes flow of energy without net flow of mass

• Flow of power P through area A is the acoustic intensity I=P/A (W m^-2)

• Pressure p and particle velocity v are related by impedance Z, and intensity I is given by product of two:

p(r,t) = Z(r) v(r,t)

instantaneous intensity I_inst(r,t) = p(r,t) v(r,t)

acoustic intensity I(r) = p_rms(r) v_rms(r) = p(r)²_max/2Z(r) = &c.

(cf. voltage and current in electronics!)

• Using phasors to represent p, v, Z may be complex (again, cf. electronics)!

• For planar waves only:

acoustic impedance Z = characteristic impedance of medium ρc

• Intensity I=P/A flows at speed c. Hence energy density E = I/c (J m^-3)

Propagation in tissue

• Is tissue a solid or a liquid?

• Can it support shear waves?

• What is propagation speed c in tissue?

Longitudal speeds of sound

• Hard tissue (bones, teeth); c ≈ 4000 m/s

• Soft tissue (muscle, fat); c ≈ 1540 m/s

• Liquid tissue (blood, lymph); c ≈ 1570 m/s

• Gas pockets (lungs, oesophagus); c ≈ 330 m/s

• compare with steel, water and air – at 37°C!

Soft tissue

• Aqueous solution with suspension of cells and matrix of extracellular scaffolding (collagen, elastin)

• Modelled as a viscoelastic gel

• Solid-like elasticity and liquid-like viscosity both contribute to presence of shear waves (~3 m/s [McLaughlin and Renzi 2006])

2. Binaural hearing

• f<2000 Hz: low α, high diffraction: interaural time difference

• f>4000 Hz: high α, low diffraction: interaural level difference

• Passive method without temporal reference

• Time differences of arrival (TDoA) in diagnostic ultrasound?

• What are diffraction and attenuation?

• Role of attenuation and diffraction in diagnostic ultrasond?

1000 Hz 1 ms ~34 cm

frequency f, attenuation coefficient α 20

cm

[Gyöngy 2010]

TDoA in medical ultrasound:

Tracking popping bubbles – passive cavitation mapping

• Cavitation (bubble activity) often involved in ultrasound therapies

• Cavitation may occur at any time (no temporal reference)

• Time differences of arrival of shockwaves allows localisation of bubbles

f>4000 Hz

Diffraction

• Point source spreads spherically

• Set of point sources interfere with each other

• Continuous source region diffracts

- analogous to interference of infinite point sources

Diffraction

• Consider single frequency f

• Pressure field p(r) expressed as complex scalar field of phasors

• Small distances |r| (near-field): p(r) = complex interference pattern

• Large distances |r| (far-field): |p| = H( θ )/|r|

• Transition on longitudal axis at |r|=D

f/4c

• Transition depends on aperture D as well as frequency!

longitudal axis aperture D

Diffraction in diagnostic ultrasound

• Typical abdominal 1D array: the L10-5 from Zonare medical systems

• Focusing in imaging plane using acoustic lens

• z=17.5 mm elevational focus

• z=60–100mm: roughly constant,

~10 mm sensitivity in elevational direction

• scattering object 5 mm out of

imaging plane may be seen!

Attenuation

• Consider a planar wave travelling in the z-direction

• Without any attenuation, the wave will maintain its amplitude:

p=A g(t-z/c)

• In reality, some of wave redirected in other direction

(scattering) and some is converted to microscopic random motion – heat (absorption)

• If attenuation is uniform over distance:

p=A exp(-αz) g(t-z/c) where α is attenuation coefficient in Nepers

• What if attenuation is caused by a single object?

Attenuation in diagnostic ultrasound

• For plane wave travelling in z-direction, attenuation

coefficient α describes “weakening” of pressure with distance:

p=A exp{j(kz-ωt)}exp(-αz)

|p|=A exp(-αz)

where α is in Nepers (Np for short).

• For tissue, α

≈ 1 dB/cm/MHz

• Therefore, at 6 MHz

- pressure amplitude halves for every cm travelled

- pressure received from perfectly reflecting target 10 cm deep (consider two-way propagation)?

• Exercise: show that 1 dB ≈ 0.115 Np

• What is origin of attenuation?

• Active method: time of transmission acts as reference

• Two-way travel time, speed of sound (SoS) → localization

• Analogy with diagnostic ultrasound?

• How accurate is the localization?

• How do echoes form from the fish (scattering)?

c ≈ 1500 m/s f = 50-200 kHz λ ≈ 3-0.75 cm

20 cm×5cm×1cm

3. Echolocation

Diagnostic echolocation:

pulse-echo B-mode imaging

• Most widespread form of diagnostic ultrasound imaging

• Very simple conceptually:

1. transmit pulse along different lines

2. convert timeline of recorded echoes to distance (d=t/2c)

3. convert amplitude of echoes to brightness on a screen

array array

Transmit and receive beamforming along thin ‘line’

Received data amplitude

(A-line)

A-line envelope

Multiple A-line envelopes create B-mode

image (here, porcine

liver in water imaged using

Localisation accuracy

Determined by width of transmit pulse autocorrelation

0 10 20 30 40

-1 0 1

pulse trace

0 0.1 0.2 0.3 0.4 0.5

0 0.5 1

frequency spectrum

-20 -10 0 10 20

-1 0 1

autocorrelation

0 10 20 30 40

-1 0 1

0 0.1 0.2 0.3 0.4 0.5

0 0.005 0.01 0.015

-20 -10 0 10 20

-1 0 1

0 10 20 30 40

-1 0 1

time

0 0.1 0.2 0.3 0.4 0.5

0 0.1 0.2

frequency

-20 -10 0 10 20

-1 0 1

time lag sinusoid

sinc

chirp

-5 0 5 -1

-0.5 0 0.5 1

0 0.5 1 1.5 2

0 0.25 0.5 0.75 1

Localisation accuracy

∆t ∆f ≈1; 2.355×0.375=0.883

#oscillations ≈ f

∆t ≈ f

/∆f = Q (=1/0.375=2.667)

Approximation better for Q>>1 (underdamping)

Scattering

• Caused by inhomogeneities of the medium (variations in compressibility κ and

density ρ)

• Total pressure field

modelled as sum of incident and scattered field:

• Hence, scattering creates

“virtual sources”

source

) , ( )

, ( )

,

( t p t p t

p r =

r +

r

λ

ka ~ 1

ka<<1 ka>>1

Regimes of echo formation (scattering):

sub-wavelength scattering resonant scattering reflective scattering

“diffusive” “diffractive” “specular” (speculum, mirror)

• k = 2π/λ: angular wavenumber

• a: characteristic size of scatterer (for sphere, equals radius)

• ka: number (dimensionless): characterises scattering behaviour

Sub-wavelength scattering (ka << 1)

• Changes in compressibility κ and density ρ has different effects:

- ∆κ causes angle-independent (monopolar) scattering

- ∆ρ causes dipolar scattering equivalent to two opposing monopoles

- θ: direction relative to direction of propagation

• Amplitude of scattered pressure increases with k and a

θ ρ θ

ρ

ρ ρ

κ κ

α κ ^cos

2 1 3 } ible incompress

fixed, {

2 cos ) (

) 3 , (

0

0 0

0

= − +

+ + −

−

s s

s

s

t

p r

monopolar scattering

∆ volumetric changes

dipolar scattering

∆ momentum changes

–+

dipole ~

2 anti-phase mpoles

• Incident pressure varies over object

• Interference between scattering wavefronts at different locations causes complicated scattered field

– backscattered wavefronts from front and back of scatterer in phase

→resonance

• Mode conversion at boundary (pressure wave ↔ shear wave) also causes resonance peaks

• By definition, in far-field of scatterer, pressure amplitude varies reciprocally with distance for constant angle:

) ) (

,

( θ

r H

t

p =

Reflective scattering (ka >> 1)

•

Scatterer very large: meetings of pressure wave with object boundary independent of each other (no phase information). (In reality, if

transmitted pulse is long enough and attenuation does not extinguish a wave before it hits a new boundary, standing waves will be set up)

•

At each boundary, mismatch in characteristic acoustic impedance (=ρc) creates reflection (as well as refraction)

•

Laws of geometric acoustics used for ray tracing (cf. optics)

•

Rays describe direction of high-frequency acoustic beams that undergo negligible diffraction or interference

medium 1

medium 2 (“scatterer”) refracted ray

reflected ray

[Ye and Farmer 1996] Water Swimbladder Fish

Mass density (kg m3) 1026 1.24 1560

Bulk modulus (MPa) 2200 0.15 2600

λ=3 cm

a≈10 cm ka≈20

Fish as (resonant) scatterers

Echolocation of airborne objects

• Air-water boundary creates great impedance mismatch

• Most sound is reflected from boundary

• Quantify this?

How does a fish school scatter?

• Multiple scattering inside fish school: diffusion of sound

• School fish as bulk inhomogeneous material: reflection

• As fish (parts) made smaller

- diffusion (causing attenuation) decreases (eventually)

- fish school becomes homogeneous medium

Acoustic concepts covered so far...

and their relevance to diagnostic ultrasound

• propagation of sound: ≈1540 m/s in soft tissue

• diffraction: focussing of mm-thick beams

• reflection and refraction: organ boundaries

• scattering: cells, collagen, elastin

• attenuation: ≈1 dB/cm/MHz

Let us review these concepts again...

and provide some additional notes

Propagation of pressure waves

• Derivation of wave equation from the governing equations of acoustics:

Eqn. of state (pressure function of density): P(R)

Continuity eqn.: (mass rate of change in dV = flux in/out dV): ∂R/∂t = - ∇·(Rv) Momentum eqn. (Newton’s second law of motion): -∇P = ρ ∂v/∂t

• Assuming small pressure and density fluctuations

P = p₀ + p where p<<p; R = ρ₀ + ρ where ρ<<ρ₀

Linearised eqn. of state: p = (κρ₀)^-1ρ (compressibility κ = 1/(-V ∂p/∂V) ) Linearised continuity eqn.: ∂ρ/∂t = - ρ₀ ∇·v

Linearised momentum eqn.: -∇p = ρ₀ ∂v/∂t

• Linear wave equation hence derived

∇²p = ∇· (∇p) = - ρ₀ ∇·(∂v/∂t) = ∂ (-ρ₀∇·v)/∂t = ∂²ρ/∂t² = c^-2 ∂²p/∂t² where c = (κρ₀)^-1/2

• Derivations in linear acoustics follow from these governing equations (e.g.

formula on wave speed, acoustic impedance of a plane wave)

Non-linear propagation

• Non-linearity arises from two effects

1. Medium non-linearity: p = A(ρ/ρ₀) + B/2(ρ/ρ₀)² + ... (p non-linear function of ρ) 2. Convective non-linearity: wave transported by particle motion

• For typical materials (B/A>1), both effects cause an increase of c with p:

1. Medium non-linearity: medium less dense than expected

2. Convective non-linearity: particle with forward motion carries pressure quicker c = c₀ + βv = c₀ + (1+B/2A)v

where β is the coefficient of non-linearity (water:5.0 blood:6.3 liver:7.8 pig fat:11.1*)

• Pressure dependent wave speed causes distortion of waveform with distance

• As a result, waveform accumulates harmonics as it travels

-0.5 0 0.5 1

-0.5 0 0.5 1

-0.5 0 0.5 1

(think of a loudspeaker placed in castor oil... what would happen as you increased the frequency?)

Non-linear processes in diagnostic ultrasound

• Non-linear propagation of ultrasound introduces harmonics into the wave as it propagates towards reflector/scatterer and back towards array, the degree of non- linear propagation being highest at the highest amplitude (focus)

• Pulse-echo imaging of such harmonics is called tissue harmonic imaging

• Air bubbles are highly non-linear scatterers, scattering sound at harmonics of the incident wave (for high enough amplitudes, they will scatter sound at the

subharmonics, ultraharmonics and even in the broadband frequency range [Neppiras 1980])

• By introducing stabilised bubbles (ultrasound contrast agents) into bloodstream, perfusion can be imaged (contrast agent imaging)

• Harmonics can be recovered in several ways:

• send one pulse and extract harmonic component of echo

• send two pulses, one inverse of other, and consider difference between two echoes (pulse inversion)

Diffraction

• Huygen’s principle: each point of non-zero pressure field (such as wavefront) is itself a superposition of point sources

• But: consider a single planar source. As it spreads in two directions, the source won’t keep splitting in two!

• Modified Huygen’s principle: point sources have directivity given by obliquity factor (maximum at propagation direction)

• Application to ultrasound transducers: pressure field result of sum of (directional) point sources across transducer surface

direction of propagation

soft baffle

transducer slit

• Reflection and refraction governed by change in characteristic acoustic impedance Z=ρc across boundary.

• Ratio of pressure reflected: (Z

-Z

)/(Z

+Z

)

• Z has units of Rayls

• For planar waves, p/|v| = Z, where v is velocity field

[Kaye&Laby] Air Water Blood Bone

Z (MRayl) 4e-4 1.5 1.1 3.5–4.6

• Over 99.9% of pressure is reflected at air-water boundary!

• Refraction governed by Snell’s law: sinθ /sinθ = c /c

Attenuation in simple conceptual terms

• Ordered vibrations of a wave gradually

• re-transmitted in other directions (scattering)

• turned into unordered, random mechanical (i.e. thermal) fluctuations (absorption)

• Simple model of wave propagation: particles held together by springs

• Wave propagation due to reaction force of springs and inertia of particles

• Scattering caused by variations in particle mass and spring stiffness

• Absorption: addition (series or parallel) of dashpots to springs [Gao et al. 1996]

scattering by

scattering by stiffer springs

• Scattering from density and compressibility changes (cf. mass-spring model)

• Classical thermoviscous model: absorption arises from phase difference between p, ρ [Lighthill 2001 pp. 78-79]

p = c²ρ + δ∂ρ/∂t; leading to ∂ρ/∂t – c^-2 ∂ρ/∂t + δc^-2∂³ρ/(∂z²∂t)=0

• Such phase difference may arise from [Cobbold 2007, pp. 84-86]

• heat conduction

• viscosity

• molecular (thermal and structural) relaxation

• Scattering: diffuse to diffractive single particules (ka≤1) α_s ~ f ^2-4 predicted

• Absorption: thermoviscous model predicts α_a ~ f ² (sim. to Kelvin-Voigt model) In contrast, α_s,α_a both ~ f ^1.1-1.2 in tissue! Modify models:

• α_s : spatial auto-correlation for ∆ρ,∆κ [Sehgal and Greenleaf 1984]

Attenuation by single objects

• Consider intensity I plane wave impinging on object with cross-section (c.s.) A

• If object removes all incident intensity (“full attenuator”) , P_removed = IA

• Object with c.s. A removes e.g. half of I acts like full attenuator of c.s. A/2

• Define acoustic c.s. as equivalent c.s. of full attenuator

• Total acoustic c.s. (area) sum of attenuation c.s. and scattering c.s.

σ = σ_a + σ_s = P_removed/ I

• Differential scattering c.s.(area/solid angle)

σ_ds(θ) = P_s(θ)/I (unlike attenuation, scattering θ-dependent)

• Differential backscattering c.s. (area /solid angle) σ_dbs = σ_ds(θ=[π 0]) (arises in pulse-echo ultrasonics)

• Backscattering coefficient (area/solid angle/volume) [Cobbold 2007, p. 308]

σ_BSC = σ_ds(θ=[π 0])/V (gives “density” of scattering)

[Au et al. 2007] Modeling the detection range of fish by echolating bottlenoise dolphins and porpoises

[Brunner 2002] Ultrasound system considerations and their impact on front-end components

[Cobbold 2007] Foundations of biomedical ultrasound [Coussios 2005] Biomedical ultrasonics lecture notes

[Gao et al. 1996] Imaging of the elastic properties of tissue – a review

[Gyöngy 2010] Passive cavitation mapping for monitoring ultrasound therapy [Hill et al. 2004] Physical principles of medical ultrasonics

[Kaye and Laby] Tables of physical and chemical constants. http://www.kayelaby.npl.co.uk/

[Lighthill 2001] Waves in fluids

[McLaughlin and Renzi 2006] Shear wave speed recovery in transient elastography and supersonic imaging using propagating fronts

...

References

...

[Neppiras 1980]Acoustic cavitation

[Olympus 2006] Ultrasonic transducers technical notes. http://www.olympus- ims.com/data/File/panametrics/UT-technotes.en.pdf

[Sehgal and Greenleaf] Scattering of ultrasound by tissues [Sekuler and Blake 1994] Észlelés

[Ye and Farmer 1996] Acoustic scattering by fish in the forward direction [Wells 1999] Ultrasonic imaging of the human body